[Paper Review] Some singular curves and surfaces arising from invariants of complex reflection groups

Cédric Bonnafé|arXiv (Cornell University)|Jul 3, 2018

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper constructs highly singular projective curves and surfaces using invariants of primitive complex reflection groups in GL₃(ℂ) and GL₄(ℂ). By leveraging Magma computations on Shephard-Todd groups, it achieves new lower bounds for the maximal number of singularities—specifically, a degree-14 curve with 42 cusps (A₂), a degree-18 curve with 36 E₆ singularities, and surfaces of degrees 8, 12, and 24 with 48, 160, and 1,440 D₄ singularities—significantly improving prior lower bounds and approaching theoretical upper bounds by Sakai and Miyaoka.

ABSTRACT

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups.

Motivation & Objective

• To systematically explore singular curves and surfaces constructed from invariants of finite complex reflection subgroups in GL₃(ℂ) and GL₄(ℂ).
• To improve known lower bounds for the maximal number of singularities of specific types (A₂, E₆, D₄) on curves and surfaces of given degree.
• To investigate whether such constructions yield examples close to theoretical upper bounds established by Sakai and Miyaoka.
• To demonstrate that Miyaoka's bounds for D₄ singularities are sharp, even for non-A-type singularities.
• To provide explicit Magma code and polynomials for reproducibility and verification of results.

Proposed method

• Utilizing the Shephard-Todd classification of complex reflection groups, the author selects primitive groups G₂₄, G₂₆, G₂₈, G₂₉, and G₃₂ for analysis.
• Computing homogeneous invariants f₁,…,fₙ of the group action to generate polynomial equations defining curves and surfaces.
• Constructing pencils of curves and surfaces via linear combinations of invariants, then analyzing their singularities using algebraic geometry tools.
• Employing Magma to compute the types and multiplicities of singularities, including Milnor numbers and quotient singularity classifications.
• Applying known upper bounds from Sakai (for A₂ singularities) and Miyaoka (for D₄ singularities) to compare lower bounds from constructions.
• Validating results via explicit polynomial constructions and referencing prior work, including preprints and computational archives.

Experimental results

Research questions

• RQ1 Can invariants of complex reflection groups in GL₃(ℂ) and GL₄(ℂ) be used to construct curves and surfaces with more singularities than previously known?
• RQ2 What is the maximal number of A₂, E₆, or D₄ singularities realizable on a curve or surface of a given degree using such invariants?
• RQ3 How close do the constructed examples come to the theoretical upper bounds for D₄ singularities established by Miyaoka?
• RQ4 Do these constructions yield varieties with large automorphism groups or unusual singularity configurations not previously documented?
• RQ5 Can the Sarti dodecic surface and other known examples be shown to be defined over ℚ using group-theoretic and computational methods?

Key findings

• A degree-14 curve with 42 cusps (A₂ singularities) is constructed using the complex reflection group G₂₄, improving the known lower bound and approaching the theoretical upper bound of 55.
• A degree-18 curve with 36 singularities of type E₆ is constructed using G₂₆, representing a new example not previously documented at this degree and singularity type.
• For surfaces of degree 8, 12, and 24, the paper establishes new lower bounds of 48, 160, and 1,440 D₄ singularities respectively, significantly improving prior results.
• The bound μ_D₄(24) ≥ 1,440 is shown to be sharp, as it is close to Miyaoka’s upper bound of 1,736, indicating that Miyaoka’s bound is nearly optimal even for non-A-type singularities.
• The construction of the degree-8 surface with 48 D₄ singularities confirms that all singular points are rational, and the Sarti dodecic surface is shown to be definable over ℚ using automorphism group theory.
• The paper provides explicit Magma code and polynomials for verification, particularly for G₃₂ and other groups, supporting reproducibility and further study.